The limiting spectral distribution of large random permutation matrices

Abstract We explore the limiting spectral distribution of large-dimensional random permutation matrices, assuming the underlying population distribution possesses a general dependence structure. Let X = (x₁, , xₙ) C ^m n be an m n data matrix after self-normalization (n samples and m features), where xⱼ = (x₁₉^*, , x₌₉^*) ^*. Specifically, we generate a permutation matrix X_ by permuting the entries of xⱼ (j=1, , n) and demonstrate that the empirical spectral distribution of Bₙ = (m/n) U ₍ X _ X _^* U ₍^* weakly converges to the generalized Marčenko–Pastur distribution with probability 1, where U ₙ is a sequence of p m non-random complex matrices. The conditions we require are p/n c >0 and m/n > 0.